1 EXISTENCE AND UNIQUENESS OF HAAR MEASURE 247 Now extend I to the whole of Jf R(G) by putting 1(0) = 0 and, for each /=/i -/2 with/1?/2 in Jf*, !(/) = I(/i) - I(/2). From (14.1.5.3) it follows immediately that !(/) depends only on/and not on the choice of expression /=/i ~/2- Clearly (14.1.53) remains valid for all /;/' in jfR(G). Further- more, the original definition of I shows immediately that I(A/) = AI(/) for all /e Jf * and all real numbers A > 0; and this relation obviously extends to the case where fe JfR(G) and AeR. We may therefore conclude from (13.3.1) that I is a positive measure on G; also I is not zero and, by construction, I(y($)/) = !(/) for all fe JfR(G). In other words, we have constructed a nonzero left-invariant positive measure on G. (2) Uniqueness. Let \L (resp. v) be a nonzero left (resp. right) invariant measure on G. Then v is a left-invariant measure. We shall show that p. and v are proportional, and this will complete the proof of (14.1.5). Let/e «2TC(G) be such that n(f) ^ 0, and consider the function Df on G defined by I f(rls)dv(f). We shall show that D^ is continuous on G. In fact this will be a consequence of the following more general result : (1 4.1 .5.5) Let Gbea locally compact group, H a closed subgroup, a a measure on H, and f: G-»C a continuous mapping. Suppose that either Supp(/) is compact or Supp(a) is compact. Then the mappings f(si) da(0 and sh-> f(ts) d are continuous on G. Consider, for example, the first of these integrals. Let ^0 e G and let V0 be a compact neighborhood of s0 . Given e > 0, we have to find a neighborhood V cV0 of Jo such that for all s e V we have.|J(/(tf) -f(s0 1)) dot(t) ^ e. If K = Supp(/) is compact and L = \VJK, then - /(s0 0) rfa(0; since / is uniformly continuous (with respect to a right-invariant distance on G) (3.16,5), there exists a neighborhood W of e in G such that the relation s e W5-0 implies \f(st) —f(s0t)\ rg e/|«|(L) for all t E G, and we may take V = V0 n W^0 . If on the other hand S = Supp(a) is compact, we havebecause/i-^In( |/j) is a seminorm.